Optimal. Leaf size=38 \[ -\frac {\log \left (b+c x^2\right )}{2 b^2}+\frac {\log (x)}{b^2}+\frac {1}{2 b \left (b+c x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 266, 44} \[ -\frac {\log \left (b+c x^2\right )}{2 b^2}+\frac {\log (x)}{b^2}+\frac {1}{2 b \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^3}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {1}{x \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b^2 x}-\frac {c}{b (b+c x)^2}-\frac {c}{b^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2 b \left (b+c x^2\right )}+\frac {\log (x)}{b^2}-\frac {\log \left (b+c x^2\right )}{2 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 33, normalized size = 0.87 \[ \frac {\frac {b}{b+c x^2}-\log \left (b+c x^2\right )+2 \log (x)}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 47, normalized size = 1.24 \[ -\frac {{\left (c x^{2} + b\right )} \log \left (c x^{2} + b\right ) - 2 \, {\left (c x^{2} + b\right )} \log \relax (x) - b}{2 \, {\left (b^{2} c x^{2} + b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 36, normalized size = 0.95 \[ -\frac {\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac {\log \left ({\left | x \right |}\right )}{b^{2}} + \frac {1}{2 \, {\left (c x^{2} + b\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 35, normalized size = 0.92 \[ \frac {1}{2 \left (c \,x^{2}+b \right ) b}+\frac {\ln \relax (x )}{b^{2}}-\frac {\ln \left (c \,x^{2}+b \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.30, size = 37, normalized size = 0.97 \[ \frac {1}{2 \, {\left (b c x^{2} + b^{2}\right )}} - \frac {\log \left (c x^{2} + b\right )}{2 \, b^{2}} + \frac {\log \left (x^{2}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.18, size = 34, normalized size = 0.89 \[ \frac {\ln \relax (x)}{b^2}+\frac {1}{2\,b\,\left (c\,x^2+b\right )}-\frac {\ln \left (c\,x^2+b\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.34, size = 34, normalized size = 0.89 \[ \frac {1}{2 b^{2} + 2 b c x^{2}} + \frac {\log {\relax (x )}}{b^{2}} - \frac {\log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________